A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866915059020595200 |
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| author | Qin, Hong |
| author_facet | Qin, Hong |
| contents | Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with time-dependent coefficients can be parameterized by an envelope matrix $w(t)$, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable iff the envelope matrix equation admits a solution with periodic $\sqrt{w^{\dagger}w}$ and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are time-dependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with time-dependent coefficients, and are expected to be effective in other studies as well, such as those on quantum algorithms for classical Hamiltonian systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_03971 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients Qin, Hong Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems Accelerator Physics Plasma Physics Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with time-dependent coefficients can be parameterized by an envelope matrix $w(t)$, which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable iff the envelope matrix equation admits a solution with periodic $\sqrt{w^{\dagger}w}$ and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are time-dependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with time-dependent coefficients, and are expected to be effective in other studies as well, such as those on quantum algorithms for classical Hamiltonian systems. |
| title | A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients |
| topic | Mathematical Physics Dynamical Systems Exactly Solvable and Integrable Systems Accelerator Physics Plasma Physics |
| url | https://arxiv.org/abs/1810.03971 |